Problem: Let $g$ be a twice differentiable function, and let $g(-5)=-6$, $g'(-5)=0$, and $g''(-5)=0$. What occurs in the graph of $g$ at the point $(-5,-6)$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $(-5,-6)$ is a minimum point. (Choice B) B $(-5,-6)$ is a maximum point. (Choice C) C There's not enough information to tell.
Solution: Since $g'(-5)=0$, we know that $x=-5$ is a critical point. The second derivative test allows us to analyze what happens in the graph of $g$ at this point according to these three cases: If $g''(-5)>0$, the graph of $g$ has a minimum point at $x=-5$. If $g''(-5)<0$, the graph of $g$ has a maximum point at $x=-5$. If $g''(-5)=0$, the test is inconclusive. [Why is this so?] We are given that $g''(-5)=0$. The test is inconclusive. There's not enough information to tell whether $(-5,-6)$ is a minimum point, a maximum point, or neither.